中文摘要：

讨论了一类带有扩散和非单调响应函数——HollingⅣ型函数的捕食模型,其中边界条件为齐次Dirichlet边界条件.首先,将该问题等价为强耦合的椭圆型边值问题,利用最大值原理和上下解方法得到正解的先验估计.然后,将该椭圆型方程组转化为一个全连续算子,利用锥上的拓扑度理论,给出正解存在的充分条件.结果表明,当食物具有群体防御能力或者猎物出现厌食时,在一定的条件下,食物和猎物可以共存.

英文摘要：

A predator-prey model with diffusion and a non-monotonic functional response,the Holling type-Ⅳ function,is discussed under homogeneous Dirichlet boundary conditions.First,the problem is equivalent to a strongly coupled elliptic boundary value problem and a priori estimate of positive solutions is deduced by means of the maximum principle and the upper and lower solution method.Then by changing the elliptic equations into a completely continuous operator and by combining with the topological degree theory in cones,sufficient conditions for the existence of positive solutions are given.Results show that the predator and prey can coexist when the prey has the ability of group defense or when anorexia response occurs on the predator population.